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Full text of "Scientific Papers - Vi"

18                                               HYDEODYNAMICAL  NOTES
When a = 27r, the two bounding radii vectores coincide and the cor vessel becomes a circle with a single partition wall at & = 0. In tl again the leading term is irrotational, being
i   ;                 &"        i     .       .. n
•fyjco = - -— 7-2 sm §6............•............
Steady Motion in a Corner of a  Viscous Fluid.
Here again we suppose the fluid to be incompressible and to move dimensions free from external forces, or at any rate from such as be derived from a potential. If in the same notation as before ^ re] the stream-function, the general equation to be satisfied by •x/r is
with the conditions that when 6 = 0 and 9 = a,
It is worthy of remark that the problem is analytically the same as a plane elastic plate clamped at 6 = 0 and 6 = a, upon which (in the considered) no external forces act.
The general problem thus represented is one of great difficulty, that will be attempted here is the consideration of one or two pa cases. We inquire what solutions are possible such that T^>, as a i of r (the radius vector), is proportional to rm. Introducing this sup-into (1), we get
as the equation determining the dependence on $.    The most gen en of i|r' consistent with our suppositions is thus
•fy - rm {A cos md + B sin mff + 0 cos (m-ty0 + D sin (m - 2) 0}, where A, B,G, D are constants.
Equation (4) may be adapted to our purpose by taking
m = mr/a,     ..............................
where n is an integer.    Conditions (2) then give
V «micircle, and the leading term (n = 1) is